mapdh8aa - Mathbox for Norm Megill

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Theorem mapdh8aa 41763

Description: Part of Part (8) in [Baer] p. 48. (Contributed by NM, 12-May-2015.)

**Hypotheses**
Ref	Expression
mapdh8a.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdh8a.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdh8a.v	⊢ 𝑉 = (Base‘𝑈)
mapdh8a.s	⊢ − = (-_g‘𝑈)
mapdh8a.o	⊢ 0 = (0_g‘𝑈)
mapdh8a.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdh8a.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdh8a.d	⊢ 𝐷 = (Base‘𝐶)
mapdh8a.r	⊢ 𝑅 = (-_g‘𝐶)
mapdh8a.q	⊢ 𝑄 = (0_g‘𝐶)
mapdh8a.j	⊢ 𝐽 = (LSpan‘𝐶)
mapdh8a.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdh8a.i	⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))
mapdh8a.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdh8aa.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
mapdh8aa.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
mapdh8aa.eg	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
mapdh8aa.ee	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)
mapdh8aa.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
mapdh8aa.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdh8aa.z	⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
mapdh8aa.zt	⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇}))
mapdh8aa.t	⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))
mapdh8aa.yn	⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇}))
mapdh8aa.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))

**Assertion**
Ref	Expression
mapdh8aa	⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑍, 𝐸, 𝑇⟩))

Distinct variable groups: 𝑥,ℎ, − 0 ,ℎ,𝑥 𝐶,ℎ 𝐷,ℎ,𝑥 ℎ,𝐹,𝑥 ℎ,𝐼 ℎ,𝐺,𝑥 ℎ,𝐽,𝑥 ℎ,𝑀,𝑥 ℎ,𝑁,𝑥 𝜑,ℎ 𝑅,ℎ,𝑥 𝑥,𝑄 𝑇,ℎ,𝑥 𝑈,ℎ ℎ,𝑋,𝑥 ℎ,𝑌,𝑥 ℎ,𝐸,𝑥 ℎ,𝑍,𝑥 Allowed substitution hints: 𝜑(𝑥) 𝐶(𝑥) 𝑄(ℎ) 𝑈(𝑥) 𝐻(𝑥,ℎ) 𝐼(𝑥) 𝐾(𝑥,ℎ) 𝑉(𝑥,ℎ) 𝑊(𝑥,ℎ)

**Proof of Theorem mapdh8aa**
Step	Hyp	Ref	Expression
1		mapdh8a.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdh8a.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		mapdh8a.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		mapdh8a.s	. . 3 ⊢ − = (-_g‘𝑈)
5		mapdh8a.o	. . 3 ⊢ 0 = (0_g‘𝑈)
6		mapdh8a.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
7		mapdh8a.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		mapdh8a.d	. . 3 ⊢ 𝐷 = (Base‘𝐶)
9		mapdh8a.r	. . 3 ⊢ 𝑅 = (-_g‘𝐶)
10		mapdh8a.q	. . 3 ⊢ 𝑄 = (0_g‘𝐶)
11		mapdh8a.j	. . 3 ⊢ 𝐽 = (LSpan‘𝐶)
12		mapdh8a.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
13		mapdh8a.i	. . 3 ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))
14		mapdh8a.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		mapdh8aa.eg	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
16		mapdh8aa.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
17		mapdh8aa.mn	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
18		mapdh8aa.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
19		mapdh8aa.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
20	19	eldifad 3923	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
21	1, 2, 14	dvhlvec 41096	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec)
22	18	eldifad 3923	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
23		mapdh8aa.z	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
24	23	eldifad 3923	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
25		mapdh8aa.xn	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
26	3, 6, 21, 22, 20, 24, 25	lspindpi 21074	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
27	26	simpld 494	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
28	10, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 16, 17, 18, 20, 27	mapdhcl 41714	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
29	15, 28	eqeltrrd 2829	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐷)
30	10, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 16, 17, 18, 19, 29, 27	mapdheq 41715	. . . . 5 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))))
31	15, 30	mpbid 232	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)})))
32	31	simpld 494	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))
33		mapdh8aa.ee	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)
34		mapdh8aa.t	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))
35	34	eldifad 3923	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
36		mapdh8aa.yn	. . . . . 6 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇}))
37	3, 6, 21, 20, 24, 35, 36	lspindpi 21074	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}) ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇})))
38	37	simpld 494	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
39	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 15, 33, 38, 25, 18, 19, 23	mapdh75d 41741	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐸)
40		mapdh8aa.zt	. . 3 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇}))
41	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 29, 32, 39, 19, 23, 40, 34, 36	mapdh8a 41762	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑍, 𝐸, 𝑇⟩) = (𝐼‘⟨𝑌, 𝐺, 𝑇⟩))
42	41	eqcomd 2735	1 ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑍, 𝐸, 𝑇⟩))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 Vcvv 3444 ∖ cdif 3908 ifcif 4484 {csn 4585 {cpr 4587 ⟨cotp 4593 ↦ cmpt 5183 ‘cfv 6499 ℩crio 7325 (class class class)co 7369 1^st c1st 7945 2^nd c2nd 7946 Basecbs 17155 0_gc0g 17378 -_gcsg 18849 LSpanclspn 20909 HLchlt 39336 LHypclh 39971 DVecHcdvh 41065 LCDualclcd 41573 mapdcmpd 41611

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-riotaBAD 38939

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-tpos 8182 df-undef 8229 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-n0 12419 df-z 12506 df-uz 12770 df-fz 13445 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-0g 17380 df-mre 17523 df-mrc 17524 df-acs 17526 df-proset 18235 df-poset 18254 df-plt 18269 df-lub 18285 df-glb 18286 df-join 18287 df-meet 18288 df-p0 18364 df-p1 18365 df-lat 18373 df-clat 18440 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-grp 18850 df-minusg 18851 df-sbg 18852 df-subg 19037 df-cntz 19231 df-oppg 19260 df-lsm 19550 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-oppr 20257 df-dvdsr 20277 df-unit 20278 df-invr 20308 df-dvr 20321 df-nzr 20433 df-rlreg 20614 df-domn 20615 df-drng 20651 df-lmod 20800 df-lss 20870 df-lsp 20910 df-lvec 21042 df-lsatoms 38962 df-lshyp 38963 df-lcv 39005 df-lfl 39044 df-lkr 39072 df-ldual 39110 df-oposet 39162 df-ol 39164 df-oml 39165 df-covers 39252 df-ats 39253 df-atl 39284 df-cvlat 39308 df-hlat 39337 df-llines 39485 df-lplanes 39486 df-lvols 39487 df-lines 39488 df-psubsp 39490 df-pmap 39491 df-padd 39783 df-lhyp 39975 df-laut 39976 df-ldil 40091 df-ltrn 40092 df-trl 40146 df-tgrp 40730 df-tendo 40742 df-edring 40744 df-dveca 40990 df-disoa 41016 df-dvech 41066 df-dib 41126 df-dic 41160 df-dih 41216 df-doch 41335 df-djh 41382 df-lcdual 41574 df-mapd 41612

This theorem is referenced by: mapdh8ab 41764

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